The /tmp and /var/tmp directories should have the sticky bit set. If not, ask your system administrator to set them. Setting the sticky bit will not break anything. It will just prevent people from deleting stuff that they don't own.
$ ls -ld /tmp /var/tmp
drwxrwxrwt 17 bin bin 16384 Mar 31 14:00 /tmp
drwxrwxrwt 2 bin bin 4096 Mar 31 14:00 /var/tmp
For example, Oracle creates sockets in /var/tmp/ (or /tmp) that is world readable and writeable. This is not controlled by oracle's UMASK. You definitely don't want to allow any user from deleting them!
$ ls -ltr /tmp/.oracle
total 0
srwxrwxrwx 1 oracle dba 0 Nov 06 21:45 s#1663178.1
srwxrwxrwx 1 oracle dba 0 Nov 20 10:54 sOracle_CSS_LclLstnr_localhost_0
srwxrwxrwx 1 oracle dba 0 Nov 20 14:37 s#1437882.1
srwxrwxrwx 1 oracle dba 0 Nov 20 14:43 s#954602.1